Many fungi undergo a reproductive phase in which more than one genetically distinct nuclei (from 2 separate mating types) is present within the same cytoplasm. In the Ascomycota and Basidiomycota, this phase is termed "dikaryotic", whereas in other fungal phyla the phase is "heterokaryotic." What is the difference here? Is it purely number of nuclei per cell, or does the difference depend on how many genetically distinct nuclei are present (2 or >2)?
In this sense, a heterokaryon is a general term, whereas a dikaryon is a specific term. A dikaryon and a trikaryon (although not often seen in literature) are both heterokaryons.
dikaryotic does - by definition - mean that there are exactly two nuclei in the cells, it does not say that the two nuclei are genetically distinct! heterokaryotic does also mean only one thing: the nuclei (the number is not important) are genetically distinct.
that's the reason why, for example webster, writes "heterokaryotic dikaryon".
in fact the nuclei are distinct in almost all cases, that's why some only write "dikaryon"